The Fermi-Pasta-Ulam-Tsingou chain was supposed to thermalize — distribute its energy equally among all modes. Instead, it recurred. Sixty years later, the conditions under which near-integrable systems eventually forget their initial conditions remain contested.
Patra and Flach (arXiv:2603.22806) approach the problem through Lyapunov spectra. They examine three routes toward integrability: FPUT chains approaching the harmonic limit, FPUT approaching the Toda lattice, and open-boundary Toda chains approaching the integrable fixed-boundary Toda. In each case, as the system approaches the integrable limit, thermalization slows down. The question is how.
The answer is in the scaling of the largest Lyapunov exponent and the Kolmogorov-Sinai entropy as functions of the distance from integrability. The Lyapunov spectrum — the full set of exponential divergence rates — contains more information than any single observable. Its shape reveals whether the nonintegrable perturbation acts locally (affecting a few modes) or globally (touching the entire spectrum).
The finding: all three cases behave as “Long Range Networks” of nonintegrable perturbations. The perturbation that breaks integrability doesn't concentrate on nearby modes — it reaches across the spectrum. This is why thermalization, though slow, is not merely delayed. The memory of initial conditions leaks out through a network that spans the system, not through local diffusion from mode to neighboring mode.
The structural point is that the approach to integrability determines the approach to equilibrium. The closer a system is to being exactly solvable, the slower it forgets — but the mechanism of forgetting is not a local process. The perturbation structure is as important as its magnitude.